Asymptotic distribution of the conditional regret risk for selecting good exponential populations

In this paper empirical Bayes methods are applied to construct selection ru les for the selection of all good exponential distributions. We modify the selection rule introduced and studied by Gupta and Liang [10] who proved th at the regret risk converges to zero with rate O(n(-lambda /2)),0 < <lambda > less than or equal to 2. The aim of this paper is to study the asymptotic behavior of the conditional regret risk R-n. It is shown that nR(n) tends in distribution to a linear combination of independent chi (2)-distributed random variables. As an application we give a large sample approximation fo r the probability that the conditional regret risk exceeds the Bayes risk b y a given epsilon > 0. This probability characterizes the information conta ined in the historical data.